B737 Captain reported an inflight shut down due to a fuel leak. Flight diverted and landed safely.

Narrative

Flight started with a 1 hour EDCT (Expected Departure Clearance Time). Crew started 1 engine for taxi out to save fuel. When we attempted to start the #1 engine IAW (In Accordance With) the MEL the start valve would not open. We contacted MX (Maintenance) and discovered a popped CB (Circuit Breaker). MX directed us to reset the CB and attempt a new start. The engine started normally; but we missed our flow time to ZZZ1. Based on the MEL for the start switch and the failed first start attempt; we elected to keep the engine running while we waited for our new EDCT. We contacted Dispatch to update our minimum take off fuel to accommodate the increased fuel burn on the ground. Once airborne; we noticed the computed remaining fuel at ZZZ1 kept decreasing; even with more direct routing. Approximately 1:45 into the flight we noticed a fuel imbalance light. We followed the fuel imbalance checklist; contacted dispatch via phone patch; and also talked with MX. The checklist procedures indicated a fuel leak. We shut down the engine IAW SOP with dispatch and Maintenance Control concurrence with our plan of action. We requested priority handling; and we diverted to ZZZ2 as the nearest suitable airfield. All checklists were completed and a safe landing was made in ZZZ2. After personnel made a visual inspection of the aircraft; and indicated they did not notice anything; we taxied to the gate with a vehicle following behind. Parking; shut down; and passenger deplaning were uneventful.